Digital Refocusing with Incoherent Holography

Figure 1: Light field propagation vs. electric field propagation: (a) Incoherent light propagation through free space is represented by 4D light fields (in some literature, propagation is expressed as a shearing of the 4D light field, which can be represented as a 4D convolution). Light field cameras capture 4D light fields on 2D sensors, resulting in loss of spatial resolution. (b) Coherent light wave propagation is represented by 2D electric fields. The 2D field can be propagated between parallel planes by a 2D convolution. The coherent blur encodes depth information and has broadband frequency response. This enables digital refocusing without loss of spatial resolution.

Project Description

Light field cameras allow us to digitally refocus a photograph after the time of capture. However, capturing a light field results in a significant loss in spatial resolution. In this paper, we propose incoherent holography for digital refocusing without loss of spatial resolution. The main idea is to capture 2D coherent holograms of the scene instead of the 4D light fields. The key properties of coherent light propagation are that the coherent spread function (hologram of a single point source) encodes scene depths and has a broadband spatial frequency response. These properties enable digital refocusing with 2D coherent holograms, which can be captured on sensors without loss of spatial resolution. Incoherent holography does not require illuminating the scene with high power coherent laser, making it possible to acquire holograms even for passively illuminated scenes. We provide an in-depth performance comparison between light field and incoherent holographic cameras in terms of the signal-to-noise-ratio (SNR). We show that given the same sensing resources, an incoherent holography camera outperforms light field cameras in most real world settings. We demonstrate a prototype incoherent holography camera capable of performing digital refocusing from only 3 acquired images. We show results on a variety of scenes that verify the accuracy of our theoretical analysis.

Publications

Images

Digital Refocusing with a Coherent Spread Functions (CSF):

Here we provide a simple simulation demonstrating how refocusing a hologram is carried out simply as a convolution with a 2D CSF. The scene consists of three playing cards placed at different depths. The coherent spread functions (CSF) for the cards vary in size according to their depths. The CSFs have the shape of a sinusoidal zone plate. Consequently, high spatial frequencies are preserved in the hologram even though the maximum blur diameter is nearly half the image width. In contrast, the incoherent PSFs in conventional imaging act as low-pass filters

Prototype Incoherent Holography Camera:

Our prototype camera utilizes a Michelson interferometer setup. A cube beam splitter divides incoming light from the scene into two beams. Two mirrors then reflect the beams back towards an objective lens that images the wavefront onto the sensor. One of the mirrors has a small amount of curvature to shift the focus of the beam relative to the other path. A piezo-actuated mirror creates sub-micron displacements that are used for phase shifting. The mirrors were both 25.4mm diameter and the objective lens was an 50mm focal length lens (Edmunds Optics 59-873), resulting in an aperture setting of approximately F/2. For the sensor, we used a 1/2″ monochrome 10 Mpix sensor with 1.67um pixels (Edmunds Optics 86-749). For the color filter, we used a 10nm bandwidth interference filter with center frequency 632.8nm (Thorlabs FL632.8-10).

Incoherent Holography Geometry:

The figure shows the geometric parameters of an incoherent holographic camera. A Michelson interferometer setup is unfolded and the two paths are shown independently. The key property is that each point source is divided into two point sources with slightly different radii of curvature. The two point sources are then recombined and interfere on the sensor plane, reproducing a Point Spread Function (PSF) very similar to the optical propagator for a coherent optical field (i.e. the CSF).

Incoherent Holography Capture Decomposition:

The incoherent holography PSF consists of a sum of four different PSFs. As a result, the captured image is a sum of four different blurred images. Two of the images are simply blurred by the incoherent PSF from the two point sources in the Michelson interferometry setup. The other two images are blurred by the CSF. Phase shifting is used to remove the undesired blurred images.

Noisy Simulations for Digital Refocusing with Incoherent Holography:

We show that The MTF of an incoherent holography system is inversely proportional to the maximum blur size. In this simulation, the scene consists of three playing cards placed at different depths. Gaussian noise is added to captured images with a std-deviation of &sigma=.005. The camera is focused on the front card (Depth 1). The scene depths are chosen so that the maximum blur size is 32 pixels (top row) and 64 pixels (bottom row). (a) The three input phase-shifted images. (b) The recovered holograms. The front card is in focus, and the other two cards are blurred. (c, d) Images refocused on the middle card (Depth 2) and the back card (Depth 3), respectively. Image noise is higher for the larger blur (bottom row).

RMSE for Digital Refocusing with Incoherent Holography:

We derive analytic expressions showing that The MTF of an incoherent holography system is inversely proportional to the maximum blur size. Since the MTF of is flat over the passband, this means that SNR falls off exactly in the same way, producing significantly greater SNR than a full resolution light field camera. To verify this result, we perform simulations with a variety of blur sizes and add Gaussian noise with &sigma=.005. As shown in this plot, the increase in noise is linear, where it is evident that the ratio of RMSE deblurring error is approximately equal to the ratio of blur sizes.

Results1Experimental results for digital refocusing:

A Playing cards scene. Depth range is approximately 75cm and the maximum blur size is approximately 1200 pixels. The scene was captured at full 10 Mpix resolution. The exposure time was 1/3 sec due to the amount of light blocked by the interference filter. The small pixels have a very small dynamic range, and we averaged 100 frames and used denoising software to increase the SNR to a reasonable level. The right image shows the coherent field recovered after phase shifting, which is focused on the King in the background. The center image shows the results after refocusing on the Jack in the foreground. The Jack is seen to come into crisp focus while the King becomes significantly blurred.

Results2Experimental results for digital refocusing:

A Diner scene. The depth range is approximately 50cm and the maximum blur size is approximately 600 pixels. The right image shows the coherent field recovered after phase shifting, which is focused on the diner in the background, while the fence is severely blurred. The high frequencies in the fence are still visible even with an extremely large blur size. The center image shows the the fence come into focus while the diner becomes blurred.

Acknowledgements

Oliver Cossairt and Nathan Matsuda acknowledge support through a Samsung GRO grant.